The p-h symmetry [1]} plays an important role in that it leads to the \(U(4)\)  symmetry of the projected Coulomb interaction, and helps in identifying the ground state in the strong coupling limit. The p-h transformation \(\hat{\mathcal {P}}\)  acts within a valley and is defined as \(i \mu _y\)  –the interchange of the two layers and change the sign of the top layer– followed by the in-plane inversion \(r \rightarrow - r\) . When keeping only the first order intra-layer gradient terms and only the contact inter-layer terms in the BM model, we have \(\hat{\mathcal {P}}^{\dagger } H_{BM}(k) \hat{\mathcal {P}} = - H_{BM}(-k)\) . However, \(\hat{\mathcal {P}}\)  is only an approximate symmetry as it is generally broken by the higher order gradient terms. For example, it is broken by the \(O(k^2)\)  and \(O(k\partial U)\)  terms in \(H_{intra}\) , as well as \(w_3\)  and the vector couplings \(\lambda _{i \ne 3}\)  in the inter-layer tunnelings. In order to quantify the degree of the p-h symmetry violation within the narrow bands, we define the \(2 \times 2\)  projected p-h matrix as \(\mathcal {P}_{i j}(k) = \langle \Psi _{i}(-k) | \hat{\mathcal {P}} | \Psi _{j, k} \rangle \) .
If the p-h symmetry is exact, it is expected that the matrix \(\mathcal {P}_{ij}(k)\)  is unitary for arbitrary \(k\) , and thus both the singular values, \(\lambda _1(k)\)  and \(\lambda _2(k)\) , are 1. Otherwise, \(\lambda _1(k)\)  and \(\lambda _2(k)\)  are smaller than 1, and therefore, the deviation of \(\lambda _i\)  (\(i = 1\) , 2) from 1 measures the p-h asymmetry of the Hilbert space. Fig. REF  illustrates the two singular values \(\lambda _i\)  for both models. While the narrow bands in Ref. [2]} are almost perfectly p-h symmetric, those in Ref. [3]} shows significant p-h asymmetry. As we demonstrate in the Fig. REF , the dominant source of the p-h asymmetry in the model of Ref. [3]} comes from the inter-layer contact coupling \(w_3\)  and the sub-dominant contribution comes from the gradient coupling \(\lambda \) .
<FIGURE>